Gradings by cyclic groups on classical simple Lie algebras in prime characteristics

Abstract

We classify, up to isomorphism, gradings by finite cyclic groups on classical simple Lie algebras $\mathfrak{g}$ over an algebraically closed field of arbitrary characteristic. Using the smoothness of the automorphism group scheme $\mathrm{Aut}\mathfrak{g}$ and correspondence between ${\mathbb{Z}}_m$-gradings and morphisms ${\mathbf{\mu }}_m\to \mathrm{Aut}\mathfrak{g}$, we express the classification as an orbit problem for certain Weyl-type groups. More generally, for the affine group scheme $\mathbf{G}$ associated to a semisimple algebraic group $G$ and the constant group scheme ${\mathbf{\Gamma }}_0$ associated to a subgroup ${\Gamma }_0$ of the automorphism group of the based root datum of $G$, we consider the classification of morphisms ${\mathbf{\mu }}_m\to \mathbf{G}⋊{\mathbf{\Gamma }}_0$ up to conjugation by $G⋊{\Gamma }_0$. We show that the classification in characteristic $p$ is the same as in characteristic $0$ except that, in characteristic $p$, only elements of ${\Gamma }_0$ whose order is prime to $p$ can occur. For ${\mathbb{Z}}_m$-gradings on $\mathfrak{g}$, this extends the classification by Kac coordinates to arbitrary characteristic, with the caveat that only diagram automorphisms of order prime to $p$ are allowed and, if $p=2$ or $3$, the type of $\mathrm{Aut}\mathfrak{g}$ is not always the same as the type of $\mathfrak{g}$.

36 pages, 1 figure, 2 tables